Noncompact $\mathbf{CP}^N$ as a phase space of superintegrable systems

TL;DR

The paper addresses how to describe superintegrable systems with dynamical $so(1,2)$ symmetry in a higher-dimensional, geometric setting by employing the noncompact Klein model as a phase space, i.e., a noncompact Kähler manifold with $su(1.N)$ isometries. It develops an explicit construction of conformal, oscillator-like, and Coulomb-like systems on $ ilde{oldsymbol{CP}}^N$, deriving their constants of motion from Killing potentials and organizing the dynamics into radial and angular sectors via canonical coordinates and action-angle variables. The main results include the explicit expressions for the Hamiltonians, constants of motion such as $M_{ ext{alpha} ext{beta}}$ and $R_ ext{alpha}$ (and their power-based generalizations), and a clear route to maximal superintegrability in higher dimensions. The work provides a geometrically natural framework for superintegrable and potentially supersymmetric extensions, with implications for quantization in noncanonical coordinates and connections to deformed integrable systems like Ruijsenaars–Schneider models.

Abstract

We propose the description of superintegrable models with dynamical $so(1.2)$ symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the non-compact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the $su(N.1)$ isometries of the Kähler structure.